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 A093883 Product of all possible sums of two distinct numbers taken from among first n natural numbers. 115
 1, 3, 60, 12600, 38102400, 2112397056000, 2609908810629120000, 84645606509847871488000000, 82967862872337478796810649600000000, 2781259372192376861719959017613164544000000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS From Clark Kimberling, Jan 02 2013: (Start) Each term divides its successor, as in A006963, and by the corresponding superfactorial, A000178(n), as in A203469. Abbreviate "Vandermonde" as V. The V permanent of a set S={s(1),s(2),...,s(n)} is a product of sums s(j)+s(k) in analogy to the V determinant as a product of differences s(k)-s(j). Let D(n) and P(n) denote the V determinant and V permanent of S, and E(n) the V determinant of the numbers s(1)^2, s(2)^2, ..., s(n)^2; then P(n) = E(n)/D(n). This is one of many divisibility properties associated with V determinants and permanents. Another is that if S consists of distinct positive integers, then D(n) divides D(n+1) and P(n) divides P(n+1). Guide to related sequences: ... s(n).............. D(n)....... P(n) n................. A000178.... (this) n+1............... A000178.... A203470 n+2............... A000178.... A203472 n^2............... A202768.... A203475 2^(n-1)........... A203303.... A203477 2^n-1............. A203305.... A203479 n!................ A203306.... A203482 n(n+1)/2.......... A203309.... A203511 Fibonacci(n+1).... A203311.... A203518 prime(n).......... A080358.... A203521 odd prime(n)...... A203315.... A203524 nonprime(n)....... A203415.... A203527 composite(n)...... A203418.... A203530 2n-1.............. A108400.... A203516 n+floor(n/2)...... A203430 n+floor[(n+1)/2].. A203433 1/n............... A203421 1/(n+1)........... A203422 1/(2n)............ A203424 1/(2n+2).......... A203426 1/(3n)............ A203428 Generalizing, suppose that f(x,y) is a function of two variables and S=(s(1),s(2),...s(n)). The phrase, "Vandermonde sequence using f(x,y) applied to S" means the sequence a(n) whose n-th term is the product f(s(j,k)) : 1<=j mul(mul(i+j, i=1..j-1), j=2..n): seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017 MATHEMATICA f[n_] := Product[(j + k), {k, 2, n}, {j, 1, k - 1}]; Array[f, 10] (* Robert G. Wilson v, Jan 08 2013 *) PROG (PARI) A093883(n)=prod(i=1, n, (2*i-1)!/i!)  \\ M. F. Hasler, Nov 02 2012 CROSSREFS Cf. A006963, A093884, A203469. Sequence in context: A165626 A120307 A022915 * A203518 A297562 A128075 Adjacent sequences:  A093880 A093881 A093882 * A093884 A093885 A093886 KEYWORD nonn AUTHOR Amarnath Murthy, Apr 22 2004 EXTENSIONS More terms from Vladeta Jovovic, May 27 2004 STATUS approved

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Last modified May 18 12:45 EDT 2021. Contains 343995 sequences. (Running on oeis4.)